Let $x$ and $y$ be two 2D vectors with their magnitudes $|x|$ and $|y|$.

Now, we take $y$ and compute the length of its projection onto $x$, which we call $y_p$:

$$y_p=|y|\cos \theta$$

Once we have the length of the projection, the dot product is:

$$x\cdot y=|x||y|\cos \theta$$

Or:

There is another way:
$$\left[\begin{array}{c}{x}_1\\ x_2\\ x_3\end{array}\right]\cdot \left[\begin{array}{c}{y}_1\\ y_2\\ y_3\end{array}\right]=x_1{y}_1+x_2{y}_2+x_3{y}_4$$

Why are those two ways the same thing?

BECAUSE, imagine if we use a tasformation with only one dimension for 2D vectors.

$$t=\left[\begin{array}{cc}2& 1\end{array}\right]$$

Now, with this transformation, the span of the system becomes a line, because both unit vectors are on the same dimension.

So if we put any vector through this transformation, it will go along this line.

Since the components of our vector are adding value to the same dimension, for us to obtain our output vector, we just need to sum its components:

So the new vector/scalar is -2.
And we can see that applying the transformation to the vector is the same operation that we described before.
In the first way, we are taking the norm/magnitudes of each vector and comparing them in one dimension.
In the second way we are collapsing everything in one single dimension(the one of the first vector), then the other vector will automatically project onto the dimension.

TO BE REVISED!
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